No. I would not support this. What's a certified trainer? Certified by whom? What organization? What method of training would be used? After the $2,000 in fines, the person gives the dog to a shelter, where it will eventually be euthanized if not adopted, so how does the dog win?

If they do, it could be because they want to feel more like part of the family, or it's just a particular animal's personality and that animal was matched with the person meant for it.

Probably but since it was a gift of $10,000 or less, it is nontaxable to you and them. Hopefully, you got a piece of paper stating that it was a gift in case later IRS wants to know where you got it.